W14 Notes

Polar curves

01 Theory - Polar points, polar curves

Polar coordinates are pairs of numbers which identify points in the plane in terms of distance to origin and angle from -axis:

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Converting

Polar coordinates have many redundancies: unlike Cartesian which are unique!

  • For example:
    • And therefore also (negative can happen)
  • For example: for every
  • For example: for any

Polar coordinates cannot be added: they are not vector components!

  • For example
  • Whereas Cartesian coordinates can be added:

⚠️ The transition formulas require careful choice of .

  • The standard definition of sometimes gives wrong
    • This is because it uses the restricted domain ; the polar interpretation is: only points in Quadrant I and Quadrant IV (SAFE QUADRANTS)
  • Therefore: check signs of and to see which quadrant, maybe need -correction!
    • Quadrant I or IV: polar angle is
    • polar angle is

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Equations (as well as points) can also be converted to polar.

For , look for cancellation from .

For , try to keep inside of trig functions.

  • For example: ⨠⨠

02 Illustration

Converting to polar: -correction

Converting to polar: -correction

Compute the polar coordinates of and of .

Solution

For we observe first that it lies in Quadrant II.

Next compute:

⨠⨠⨠⨠

This angle is in Quadrant IV. We add to get the polar angle in Quadrant II:

⨠⨠

The radius is of course since this point lies on the unit circle. Therefore polar coordinates are .

For we observe first that it lies in Quadrant IV.

Next compute:

⨠⨠⨠⨠

This is the correct angle because Quadrant IV is SAFE. So the point in polar is .

Shifted circle in polar

Shifted circle in polar

For example, let’s convert a shifted circle to polar. Say we have the Cartesian equation:

Then to find the polar we substitute and and simplify:

⨠⨠⨠⨠⨠⨠⨠⨠⨠⨠

So this shifted circle is the polar graph of the polar function .

03 Theory - Polar limaçons

To draw the polar graph of some function, it can help to first draw the Cartesian graph of the function. (In other words, set and , and draw the usual graph.) By tracing through the points on the Cartesian graph, one can visualize the trajectory of the polar graph.

This Cartesian graph may be called a graphing tool for the polar graph.

A limaçon is the polar graph of .

Any limaçon shape can be obtained by adjusting in this function (and rescaling):

Limaçon satisfying : unit circle.

Limaçon satisfying : ‘outer loop’ circle with ‘dimple’:

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Limaçon satisfying : ‘cardioid’ ‘outer loop’ circle with ‘dimple’ that creates a cusp:

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Limaçon satisfying : ‘dimple’ pushes past cusp to create ‘inner loop’:

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Limaçon satisfying : ‘inner loop’ only, no outer loop exists:

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Limaçon satisfying : ‘inner loop’ and ‘outer loop’ and rotated :

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Transitions between limaçon types, :

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Notice the transition points at and :

The flat spot occurs when

  • Smaller gives convex shape

The cusp occurs when

  • Smaller gives dimple (assuming )
  • Larger gives inner loop

04 Theory - Polar roses